Compression behavior of loose wet granular materials experiment and discrete numerical simulation

The behavior under compression, either oedometric or isotropic, of model granular terials glass beads wet by a small quantity of liquid forming capillary bridges is studiedboth by discre

ÉCOLE DOCTORALE SCIENCE INGÉNIERIE ET ENVIRONNEMENT

THÈSEprésentée pour l'obtention du diplơme docteur de

L'UNIVERSITÉ PARIS-ESTSpécialité: Géotechnique

parVinh Du THAN

Sujet de la thèse :

Compression behavior of loose wet granular materials:

experiment and discrete numerical simulation

Thèse soutenue le 26 avril 2017 devant le jury composé de :

Gặl COMBE Professeur, Université Grenoble Alpes Président

Jean-Yves DELENNE Directeur de recherche, INRA Rapporteur

Bertrand FRANÇOIS Professeur, Université Libre de Bruxelles Rapporteur

Jean-Michel PEREIRA Chercheur, École des Ponts ParisTech ExaminateurAnh Minh TANG Chercheur, École des Ponts ParisTech Directeur de thèse

Ce travail de thèse n'aurait être possible sans appui de beaucoup de gens Je tiens àremercier dans cette page un certain nombre de peronnes qui m'ont aidé à mener à bien

ce travail

Je tiens tout d'abord à remercier Monsieur Gặl Combe, qui m'a fait l'honneur deprésider le jury de ma soutenance de thèse ainsi qu'a examiné mon travail Ses remarques

et questions sont très intéressantes et constructives

Je voudrais remercier vivement Messieurs Jean-Yves Delenne et Bertrand Françoisqui ont accepté la longue et lourde tâche d'être rapporteurs de ma thèse Leurs remarquesfructueuses et présieuses m'ont aidé à améliorer le niveau scientique de ce travail

Je souhaite exprimer ma profonde gratitude à mes trois directeurs de thèse AnhMinh Tang, Jean-Noël Roux et Jean-Michel Pereira pour leur conance, leur suivi et leurgentilesse Merci à Jean-Michel pour tes conseils précieux et ton aide Merci sincèrement

à Jean-Noël pour ton modèle de la simulation, tes conseils précieux, tes encouragementsainsi que ta patience dans les moments que je n'ai pas pu bien comprendre ton modèle

Si ce travail est une réussite, c'est une très grande partie grâce à toi Un grand merci

à Minh pour avoir accepté ma demande à faire ma thèse à l'École des Ponts ParisTech.C'est un grand honneur pour moi Tu m'as beaucoup aidé, tu m'as guidé et accompagnétout le long de l'élaboration de ce travail Tes soutiens et encouragements m'ont permismener à bien ce travail

Je voudrais également adresser mes remerciements aux Messieurs Michel Bornert etPatrick Aimedieu de l'équipe multi-échelles, qui m'ont aidé de faire des essais de micro-tomographie aux rayons X Leurs conseils sont très utiles pour la partie du traitementd'imagerie

Je tiens aussi à remercier l'équipe technique du CERMES: Emmanuel, Hocine, Xavier,Baptiste, Marine pour leurs aides à faire des essais en laboratoire

Je n'oublie pas, non plus, de remercier le Ministère de l'Éducation et la Formation

du Vietnam pour avoir nancé cette thèse ainsi que les soutiens de mes collègues dans

le département de génie civile de l'Institut Universitaire de Technologie de l'Université

de Danang

J'aimerais remercier tous les chercheurs et les amis dans l'équipe CERMES pour

i

l'ambiance de travail et les très bons moments qu'on a eu ensemble Mes remerciementsvont ensuite à mes amis vietnamiens avec qui j'ai partagé des moments inoubliables dans

la vie quotidienne

J'aurais une grande reconnaissance à mes parents, mes grands frères, ma grande s÷ur

et ma petite s÷ur pour leurs soutiens et encouragements

Enn, j'aimerais remercier inniment mon épouse Phuong Thao et mon ls VinhKhoa qui ont fait tant de sacrices pour moi avec l'amour et un soutien sans limites Àeux, je dédie cette thèse

Champs-sur-Marne, le 10 mai 2017

THÂN Vinh Du

cation and Training of Vietnam (project 911).

Ce travail a été réalisé au Laboratoire Navier de l'Ecole des Ponts ParisTech, versité Paris-Est La bourse de thèse est nancée par le Ministère de l'Education et laFormation du Vietnam (projet 911)

Uni-iii

The behavior under compression, either oedometric or isotropic, of model granular terials (glass beads) wet by a small quantity of liquid forming capillary bridges is studiedboth by discrete element numerical simulation (DEM) and by laboratory experiments,which combine oedometric tests and X-ray microtomography (XRCT) Special attention

ma-is paid to very loose initial states (solid fraction below 30 %) stabilized by capillarycohesion XRCT observations involve suitable spherical particle detection adapted torelatively low resolution images It enables heterogeneities to be visualized and microsc-tructural information to be collected DEM simulations lead to the identication ofparameters that inuence the plastic compression behavior of the material Importantfactors include in particular the initial coordination number of the loose contact network,which varies to a large extent as a consequence of the ability of the assembling proce-dure to restructure cohesive clusters in their aggregation stage; and some possible slightresistance to rolling and pivoting in the contacts The macroscopic compression curve(the void ratio dependence on stress) is controlled by a dimensionless reduced pressurecomparing conning stress to capillary forces In the stress interval corresponding toirreversible collapse, it assumes the classical logarithmic form of soil mechanics Forthis compression law as well as for various microscopic observations, experiments andsimulations are found to agree semi-quantitatively

Keywords: DEM, wet granular materials, one-dimensional compression,isotropic compression, X-ray computed tomography, grain-scale analysis, plas-tic behavior, microstructure

v

Le comportement en compression, oedométrique ou isotrope, de matériaux granulaires(billes de verre) en présence d'une faible quantité d'eau formant des ponts capillairesest étudié par la simulation numérique aux éléments discrets (DEM) et par des expéri-ences en laboratoire, combinant essais oedométriques et microtomographie aux rayons X(XRCT) On porte une attention particulière aux états initiaux très lâches (compacitéinférieure à 30 %) stabilisés par la cohésion capillaire L'observation par XRCT, fondéesur une méthode de détection des grains sphériques adaptée aux images relativementpeu résolues, permet de visualiser les hétérogénéités et d'accéder à des informationsmicrostructurelles L'approche numérique par DEM permet de dégager les paramètresinuençant le comportement de compression plastique du matériau Parmi ceux-ci lacoordinence initiale du réseau des contacts, plus ou moins élevée selon que le procédéd'assemblage permet ou non aux amas connectés par la cohésion capillaire de se restruc-turer, ainsi qu'une éventuelle légère résistance au roulement ou au pivotement dans lescontacts, jouent un rôle déterminant La courbe de compression macroscopique (varia-tion de l'indice des vides avec la pression) est régie par une pression réduite qui compareles eets du connement à l'attraction capillaire, et prend la forme, dans l'intervalle decontraintes correspondant à un eondrement irréversible, de la loi logarithmique familièredans les sols cohésifs Pour cette loi comme pour diérentes observations microscopiques,

un accord semi-quantitatif est obtenu entre expériences et simulations numériques

Mots-clés: Modélisation des élements discrets, matériaux granulaires lés, compression oedométrique, compression isotrope, microtomographie auxrayons X, analyse à l'échelle du grain, comportement plastique, microstruc-ture

mouil-vii

Acknowledgment i

1.1 Mechanical behavior of granular materials 7

1.1.1 General considerations 7

1.1.2 Bonded granular materials 10

1.1.3 Wet granular materials 12

1.2 Experimental investigations 15

1.2.1 Techniques 20

1.2.2 Typical observations 22

1.3 Numerical investigations 28

1.3.1 Numerical methods 28

1.3.2 Typical observations 30

1.3.3 Some characteristics of wet granular material in DEM simulation 34 1.4 Experimental investigation versus numerical investigation 39

1.4.1 Existing studies combining experimental and numerical methods 39 1.4.2 Combination between the XRCT and the DEM 43

1.5 Conclusion 49

2 Experimental investigations 51

2.1 Introduction 53

2.2 Material and methods 53

2.2.1 Material 53

2.2.2 Specimen preparation 54

2.2.3 Oedometric compression tests 57

2.2.4 X-ray computed tomography tests 59

2.3 Mechanical behavior under increasing vertical stress 62

2.4 Global scan observations 63

2.4.1 3D reconstruction from the image stack 63

2.4.2 Segmentation process 65

2.4.3 Heterogeneity 66

2.4.4 Pore scale deformation 71

2.5 Local scan observations 74

2.5.1 General principles 74

2.5.2 Algorithm to detect spherical structures 76

2.5.3 3D reconstruction after detection process 80

2.5.4 Grain size distribution after the detection process 82

2.5.5 General description of the structure at grain-scale 82

2.5.6 Heterogeneity 85

2.5.7 Geometrical structure analysis 86

2.5.8 Contact networks 92

2.6 Links between macro and microscopic behaviors 94

2.7 Conclusion 95

3 Isotropic compression: DEM study 97 3.1 Introduction 99

3.2 Model material and interaction laws 99

3.2.1 Particle shape and size distribution 99

3.2.2 Interaction law: normal forces 100

3.2.3 Tangential forces 104

3.2.4 Resistance to rolling and pivoting (RPR) 105

3.3 Basic denitions and properties 106

3.3.1 DEM equations, boundary conditions and stress control 106

3.3.2 Equilibrium condition 108

3.3.3 Coordination numbers 109

3.3.4 Liquid content and limit of pendular state 110

3.3.5 Average normal force 111

3.3.6 Dimensionless control parameters 112

3.4 Simulation procedures for the compression test 113

3.4.1 Specimen preparation 113

3.4.2 Simulation parameters 115

3.4.3 Compression cycle 118

3.5 Material behavior in isotropic compression 118

3.5.1 A reference case 118

3.5.2 Inuence of drying or of saturating 130

3.5.3 Inuence of initial state 131

3.5.4 Eects of polydispersity 138

3.6 The case of wet beads with RPR 143

3.6.1 Inuence of initial solid fraction 143

3.6.2 Inuence of initial agitation intensity 145

3.6.3 Inuence of rolling and pivoting friction coecients 149

3.6.4 Comparison of bead assemblies with and without RPR 151

3.7 Conclusion 153

4 Oedometric compression: DEM study and experimental confrontation155 4.1 Introduction 156

4.2 Simulation of oedometric compression tests 156

4.2.1 Compression of dierent initial states: initial V0 and coordination 157 4.2.2 Inuence of initial solid fraction 162

4.3 Oedometric versus isotropic compression 166

4.4 Experiment versus simulation 168

4.4.1 Compaction behavior 168

4.4.2 Geometrical structure change 170

4.5 Conclusion 171

1.1 Particles displacement during triaxial compression 9

1.2 Photoelastic image of a small assembly of disks 9

1.3 Skeletal versus contact-level capillary and electrical forces 12

1.4 Schematic view of a XRCT 21

1.5 One-dimensional compression plots for carbonate and silica sands 23

1.6 Capillary bridges in glass beads 23

1.7 Water menisci form at contacts of sand grains: (a) concave menisci; (b) convex menisci 24

1.8 The dierences of water content for dierent regions 1, 2, 3, and 4 as a function of time 25

1.9 An example of XRCT scans of Hostun sand, with D is the diameter of cell and r is the resolution of image 25

1.10 Examples of partitioning and subsequent ball-and-stick network construc-tion, applied to an unconsolidated sand imaged at 2.8 µm 26

1.11 Four dierent domains (fully saturated, funicular, pendular and hygro-scopic) shown in slices from trinarised 3D images of Hostun sand 27

1.12 Eect of soil compaction on interpores (a) and intrapores (b) volumes from soil aggregates with varying levels of soil moisture and compaction 28 1.13 The cycle of the DEM calculation 30

1.14 The packing fraction of dierent sized glass beads as a function of the eective gravitational acceleration 31

1.15 Ballistic deposit before and after compaction by a constant external force on the piston 32

1.16 Compression and decompression curves: eect of initial agitation level in aggregation stage and inuence of rolling resistance parameter 32

1.17 A map of tensile (green) and compressive (red) forces in a thin layer cut out in the packing 33

1.18 Schematic view of liquid bridge between two smooth (a) polydisperse spheres and (b) monodisperse spheres 34

1.19 Capillary force and capillary pressure as a function of the gorge's radius

for dierent values of contact angle θ 35

1.20 Evolution of the capillary force Fcap as a function of the interparticle distance h for a given capillary pressure ∆u 37

1.21 Loose cohesive assembly of disk 38

1.22 An example of compression curve of loose systems obtained with 2D and 3D models 39

1.23 Illustration of the clumping of spheres method with no overlapping particles 44 1.24 Pore size distributions of granules produced under high shear 45

1.25 Comparison of coordination measurements on numerical sphere packings produced by DEM and by analysis of XRCT images 46

1.26 Comparison of the RDFs of (a) the monodisperse glass packing structure, and (b) polydisperse glass packing structure 47

1.27 (a) Cemented numerical sample and (b) laboratory sample 48

2.1 Cumulative grain size distribution of glass beads 54

2.2 Specimen preparation for cell φ50 54

2.3 Specimen preparation for cell φ20 55

2.4 Schematic view of the system for preparing the specimen 56

2.5 Initial void ratio e0 and initial porosity n versus height of free fall hf 57

2.6 Schematic view of oedometric compression system 58

2.7 Applied force versus time during compression process The inset shows the signal noise during the initial stage when the specimen approached and touched the piston 58

2.8 Global view of in-situ compression test combined with XRCT at Labora-toire Navier 59

2.9 (a) The PMMA cell with several gaskets and pistons (heights δ given in mm, see Fig 2.10 for denition of δ) (b) Several wedges with dierent values of thickness 60

2.10 Illustration of the compressed heights of in-situ compaction test that were scanned for XRCT 61

2.11 Void ratio versus vertical stress during oedometric compression tests for cells φ50 and φ20 62

2.12 3D reconstructed images of the specimen at dierent stages of compaction 64 2.13 Illustration of grey level histograms for all global scans 65

2.14 Images at dierent steps of segmentation: (a) original image, (b) seg-mented image 66

2.15 Illustration of grey level as a function of the height of specimen 66 2.16 Schematic view of several types of diagram for calculating the heterogeneity 68

2.17 Void ratio distribution according to the compressed height of sample for

dierent compaction steps (diagram 1) 69

2.18 Void ratio distribution according to the average radii of the selected cylin-ders for dierent compaction steps (diagram 2) 69

2.19 Void ratio distribution according to eight equal sectors for dierent com-paction steps (diagram 3) 70

2.20 Void ratio distribution according to ve equal volumes for dierent com-paction steps (diagram 4) 70

2.21 Void ratio distribution according to the x (a) and y (b) directions for dierent compaction steps (diagram 5) 71

2.22 Total volume and total pore volume of volume V2 for dierent steps of compaction 72

2.23 An example of the local thickness for volume V2 (a) at step S01G and a slice (b) extracted from (a) 72

2.24 Distribution of pore volume of the volume V2 for dierent levels of com-paction 73

2.25 Position of the local scan (b) from the global scan (a) and an example of investigated cube (c) 74

2.26 3D reconstructed image of four local scans 75

2.27 The algorithm to detect the spherical structures 76

2.28 Construction of the accumulation array from the gradient eld 77

2.29 Denition of the signature curve 78

2.30 An example of a signature curve computed from the 3D image 79

2.31 A slice in a 3D detected image 79

2.32 Method to nd the lost contacts, inside cube is the standard volume; outside cube is the extended volume 80

2.33 Example of 3D reconstructed cubes of four local scans Left-hand side: before detection, right-hand side: after detection 81

2.34 Grain size distribution of the 3D reconstructed specimens 82

2.35 Example of 3D reconstructed specimen 84

2.36 Number of particles of the ten SVs for four local scans after detection process 85

2.37 Macroscopic solid fraction (triangle) and the average solid fraction of 4 local scans before and after the detection process 86

2.38 Average total coordination numbers of 4 local scan with three values of tolerance (0, 1 and 2 voxels) 87

2.39 Coordination numbers of close neighbors for four local scans 88

2.40 Number of contacts per grain for three tolerances of 0, 1 and 2 voxels 89

2.41 An example of a sphere with one contact 90

2.42 Radial distribution functions for particle centers of four local scans 91

2.43 Contact network of a typical SV under growing of compaction 93

2.44 3d-slices of the typical SV under growing of compaction 94

3.1 Grain size distribution of the polydisperse system 100

3.2 Static normal force, FN = Fe N + Fcap, as a function of the interparticle distance h 102

3.3 Coulomb cone limiting the value of the tangential force 104

3.4 Motion of two spheres 107

3.5 Three spheres connected with liquid bridges 110

3.6 Capillary force law Fcap(h), for two dierent meniscus values, according to the Maugis approximation and the Soulié formula 112

3.7 Monodisperse system on FCC lattice 114

3.8 Typical specimens of (a) the monodisperse (Φ0 = 0.30) and (b) the poly-disperse (Φ0 = 0.24) systems 115

3.9 ((a) e versus log P∗ for reference system in compression cycle; (b) com-parison of reference case and typical non-cohesive behavior 119

3.10 Eect of dierent (isotropic) unloading and reloading histories on void ratio120 3.11 a) Coordination numbers z, zc and zd; and (b) coordination numbers zc, z+, and z− versus P∗ in compression and decompression cycles 121

3.12 Contact networks in loading (a, b, and c) and unloading (d, e, and f) cycle122 3.13 Networks of compressive contacts (red lines), tensile ones (green lines), and contacts with FN = 0 (magenta) 123

3.14 Distribution of normal forces for dierent values of P∗ from the com-pression path, (a) normalized by the maximum tensile force F0, and (b) normalized by the average normal force hFNi 124

3.15 Distribution of normal forces for dierent values of P∗ in non-cohesive case, normalized by the average normal force hFNi 125

3.16 Void ratio (a), coordination numbers zc and zd(b) versus P∗ of the refer-ence case for dierent values of meniscus Vm/d3 126

3.17 Coordination numbers of z+ and z− versus reduced pressure P∗ for dier-ent values of Vm/d3 127

3.18 Distribution of the number of contacts per grain of the reference case for dierent values of Vm/d3 at three typical states of P∗ 128

3.19 Evolution of coordination number of close neighbors z(h) versus dimen-sionless interparticle distance h/d for dierent values of P∗ in loading path129 3.20 Void ratio versus pressure P , as cohesive forces are suppressed at the beginning of unloading 130

3.21 (a) Void ratio e, and (b) coordination numbers zc and zd versus P∗ for dierent values of initial solid fraction Φ0 132

3.22 Coordination numbers z+ and z− versus P∗ for dierent values of Φ0 133

3.23 Distribution of the number of contacts per grain for dierent values of Φ0

at three typical states of P∗ 1343.24 Distribution of normal forces for dierent values of Φ0 under low pressure

P∗ = 10−2, is normalized by the maximum tensile force F0 1343.25 (a) Compression and decompression curves, and (b) coordination numbers

zc and zd for dierent values of V0/V∗ 1353.26 Coordination numbers z+ and z− versus P∗ for dierent values of V0/V∗ 1363.27 A 3D slice of specimen for dierent values of V0/V∗ at the beginning ofloading path (P∗ = 10−3) 1373.28 Number of contacts per grain for dierent values of V0/V∗at three typicalstates of P∗ 1383.29 Distribution of normal forces for dierent values of V0/V∗ under P∗ =

10−3, normalized by F0 1383.30 (a) Void ratio e and (b) coordination numbers zc and zd versus P∗ fordierent values of V0/V∗ 1393.31 Contacts (red lines) and distant interactions (blue lines) for dierent val-ues of V0/V∗ at P∗ = 10−3 1403.32 Grains contained in a slice (thickness 3dmin), at low P∗, for dierent values

of V0/V∗ 1413.33 (a) Void ratio e and (b) coordination numbers zc and zd versus P∗ forMDS and PDS 1423.34 (a) Void ratio e and (b) coordination numbers zc and zd versus P∗ incompression cycle for dierent values of Φ0 1433.35 Coordination numbers z+ and z− versus reduced pressure P∗ for dierentvalues of Φ0 1443.36 (a) Void ratio e and (b) coordination numbers zc and zd versus P∗ incompression cycle for dierent values of V0/V∗ 1453.37 Coordination numbers z+ and z− versus P∗ for dierent values of V0/V∗ 1463.38 3d-slices of specimens assembled with RPR for dierent values of V0/V∗ ,under P∗ = 10−3 1473.39 Number of contacts per grain for six values of V0/V∗at three typical states

P∗ = 10−3, P∗ = 10−1, and P∗ = 101 1483.40 Distribution of normal forces at low pressure P∗ = 10−3 for dierent

V0/V∗ P.D.F normalized by F0 1493.41 (a) Void ratio e and (b) coordination numbers zc and zd versus P∗ fordierent values of µR = µP 1503.42 Distribution of normal forces for dierent values of µR/dat P∗ = 10−2 1513.43 (a) Void ratio e and (b) coordination numbers zc and zd versus P∗ incompression cycle for systems both without RPR and with RPR 152

4.1 (a) Void ratio e, and (b) coordination numbers zc and zd versus σ∗

1 fordierent values of V0/V∗ 1574.2 Coordination numbers z+(compressive contacts) and z−(tensile contacts)versus reduced axial stress σ∗

1 for dierent values of V0/V∗ 1584.3 Coecient of lateral pressure K0 versus reduced vertical stress σ∗

1 fordierent values of V0/V∗ 1594.4 Contribution of distant interactions to normal stress σ22 Red arrow de-notes the increasing of values 1604.5 a) Contributions of tangential (a) and capillary (b) interactions to normalstress σ22 1614.6 (a) Void ratio e, and (b) coordination numbers zc and zd versus σ∗

1 fordierent values of initial solid fraction Φ0 1624.7 Coordination numbers of z+ and z− versus reduced vertical stress σ∗

1 fordierent values of Φ0 1634.8 K0 versus reduced axial stress σ∗

1 for dierent values of Φ0 1644.9 Contribution of distant interactions to normal stress σ22 for dierent val-ues of Φ0 1644.10 (a) Contributions of tangential (a) and capillary (b) interactions to normalstress σ22 for dierent values of Φ0 1654.11 (a) Void ratio e, and (b) coordination numbers zc and zd versus reducedmean stress σ∗

m for the minimum and maximum V0/V∗ in both simulations

of oedometric and isotropic tests 1664.12 Distribution of normal forces for both oedometric and isotropic models 1674.13 Oedometric compression curves in experiments and simulations (no RPR)for dierent values of Φ0 1694.14 Coordination numbers of neighbors z(h) versus vertical stress σ1 and thetotal coordination number z versus positions of scans 171

1.1 Schematic diagrams for classication of wet granular materials with

vari-ous amount of liquid 13

1.2 Examples of imaging techniques to porous media systems 17

1.3 Combining experimental observations with computer simulations 42

2.1 Initial parameters of selected specimens after preparation The sample CT03 was prepared for the XRCT tests 57

2.2 Details of the scanning program G: global scan, L: local scan 61

3.1 Initial parameters of the monodisperse system without RPR 116

3.2 Initial parameters of the polydisperse system without RPR 116

3.3 Initial parameters of the monodisperse system with RPR 117

3.4 Probability of meniscus formation between close neighbors versus P∗ 130

4.1 Initial parameters of the oedometric simulations (without RPR) 156

Granular materials appear in various natural and industrial processes They are posed of individual grains that vary in shape, size, composition, surface texture, etc.These granular characteristics can remarkably aect the packing and the contact dis-tribution properties of granular assemblies Therefore, granular materials may exhibitquite dierent behaviors depending on the applied external forces, the size and geom-etry of the grains, the density and composition of particles, the type of interactions atthe grain scale and many other material properties Because of these complexities, it

com-is generally dicult to relate in a straightforward manner the macroscopic properties

of granular materials to the microscopic properties of their constituents In the case ofwet granular materials, the existence of liquid menisci between particles plays a key role

in the overall behavior of the assembly Capillary cohesion bestows to these materialsspecic mechanical features that do not exist with dry grains, such as the ability to formstable structures with very low density, and a strong sensitivity to stress intensity aswell as to stress direction So far, many experimental and numerical studies of bondedgranular materials in general and wet granular materials in particular have been carriedout to tackle the diculties exposed here above

Published studies cover a wide range of applications For instance, some investigatethe mechanical properties of cohesive soils (clays and silts) [1,2,3,4,5], metallic powderprocessing [6] or modeling and treatment of ceramic powders [7,8,9] Assemblies of wetbeads were observed [10, 11, 12], and descriptions of such materials at microscopic scalewere made possible by using numerical and experimental tests [11,12] A 2D numericalinvestigation into the microstructure and mechanical properties of cohesive powders isreported in [13,14] In addition, some numerical studies of cohesive materials have shownthat the stability of loose structures formed by particles packed under gravity rely onadhesive forces (see for instance Dong et al [15]) Other numerical works studied loosepackings stabilized by cohesion and their collapse when subjected to increasing loadsduring oedometric compression [16, 17, 18, 19] Some studies focused on the failure ofbounded particle assemblies in static [20, 21] or dynamic conditions [22] Other works

on wet bead packs in which cohesion stems from liquid bridges joining neighbouringparticles investigated the structure of poured samples [23], or their shear strength [24]

1

Shear ow of cohesive granular materials has also been simulated [25, 26, 27, 28, 29].Besides, the collapsible behavior of loessic soils [30, 31, 32], as well as wet sand withcapillary bridges [33, 34, 35] has been investigated experimentally The contribution ofarticial solid bridges to the micro-macroscopic behavior of soils, especially in the case

of cemented sands, has been studied using both numerical simulations and experimentalobservations [36, 37, 38, 39, 40, 41, 42] These works evidenced the link between themacroscopic mechanical behavior and changes in the microstructure However, moststudies on wet granular materials have focused on dense states, and few works havefocused on the grain-scale behavior

Currently, the grain-scale behavior of wet granular materials is usually investigatedusing numerical simulations, such as the discrete element method [43], in 2D and/or3D [44,45,17,29,24,46,27,47] These studies characterized precisely the microstructure

in terms of coordination number of contacts, coordination number of distant interactions,coordination number of compressive and tensile bonds, radial distribution function, forcechains, distribution of forces between particles, etc The evolution of these microstruc-tural descriptors with the externally applied pressure was also investigated However, fewworks have quantitatively compared simulations (e.g with the discrete element method)with experimental results (e.g with microstructural observations using the X-ray com-puted tomography) in order to validate the numerical method [48, 49,50, 36, 37].Following the two-dimensional model for cohesive powders which is based on thediscrete element method and developed by Gilabert et al [13, 14], we propose a three-dimensional model to investigate the mechanical behavior of a wet granular material atvery loose states of density On the same material, we also perform one-dimensionalcompression tests combined with microstructural observations using X-ray computedtomography Thereby, we provide a comprehensive view of the mechanical behavior, aswell as a further insight into microstructure changes of wet granular materials at veryloose states under growing applied external forces In the present study, we only focus

on the properties of wet granular materials in the pendular state

This thesis is organized in four chapters, as follows:

In Chapter 1, the mechanical behavior of granular materials is briey described.Several typical experimental and numerical observations of wet granular materials arethen presented in details The combinations of experiments and simulations which havebeen studied in literature are also presented in order to provide a comprehensive view ofwet granular materials

In Chapter2, we present two experimental tests on wet spherical beads at very loosestates The behavior is characterized at the macroscopic scale as well as at the grain(and pore) scale, by combining one-dimensional compression tests and X-ray computedtomography

In Chapter 3, following the 2D model from Gilabert et al [13, 14], we propose a 3Dmodel to investigate the mechanical behavior of wet spherical glass beads at very loose

states under isotropic compression Three systems, including monodisperse systemswithout and with rolling and pivoting resistances and a polydisperse system withoutrolling and pivoting resistances, are simulated The plastic response of the material, aswell as the evolution of microstructure and force transmission, along compression anddecompression paths are then characterized The inuence of various micromechanicalparameters have also been investigated.

In Chapter 4, we rst propose a 3D observation of wet spherical glass beads in

an oedometrically compressed model We then validate the compaction behavior andmicrostructural changes by comparing the results of experiments and simulations.Finally, some conclusions and perspectives are drawn

Literature review

In this chapter, the mechanical behavior of granular materials is briey described,with a particular focus on wet granular materials Several non-destructive tech-niques to observe the micro-macroscopic behavior of such materials are thenpresented Numerical investigations, which provide a detailed access to the mi-crostructure of these materials, are then described A comparison of experimentaland numerical results are nally presented

Contents

1.1 Mechanical behavior of granular materials 7

1.1.1 General considerations 7

1.1.2 Bonded granular materials 10

1.1.3 Wet granular materials 12

1.3.3 Some characteristics of wet granular material in DEM simulation 34

1.4 Experimental investigation versus numerical investigation 39

1.4.1 Existing studies combining experimental and numerical methods 39

1.4.2 Combination between the XRCT and the DEM 43

5

1.5 Conclusion 49

1.1 Mechanical behavior of granular materials

1.1.1 General considerations

Granular materials are ubiquitous in various elds, and natural or industrial processesincluding soft matter physics, soil mechanics, powder technology, agronomic transfor-mations, and geological processes They are composed of individual grains that vary inshape, size, and surface texture Because of the wide variety of physicochemical andmorphological grain properties, the fundamental behavior of granular materials should

be investigated under dierent working environments The analysis of physical systemsinvolving granular materials requires a clear understanding of their behavior not only atthe single particle level, but should also consider multi-physics problems involving multi-scale phenomena from molecular to macroscopic scales A brief description of granularmaterials in general is now given

Satake and Tobita [51] dened granular materials as grains in contact and the rounding voids The micromechanical behavior of granular materials is therefore inher-ently discontinuous and heterogeneous The macroscopic (overall or averaged) behavior

sur-of granular materials is determined not only by how discrete grains are arranged in space,but also by what kinds of interactions are operating among them In order to under-stand the mechanical behavior of granular materials from a microscopic point of view,the spatial distribution and orientation of grains and their contact conditions should be

rst specied

Lanier [52] described granular materials as an intermediate class of materials between

uids and solids, that not only ow like uids (snow avalanches, emptying silos, etc.) butalso resist to deviatoric stresses (like solids) This type of materials is in the category ofcomplex structured materials, anisotropic, and strongly heterogeneous Their mechanicalbehavior depends on the interactions occurring at the particle level Noteworthy is thefact that this behavior is partially reversible only within a very restricted domain andthat it is non-linear

After Radjạ and Lanier (in Cambou et al [53]), granular materials consist of denselypacked solid particles and a pore-lling material which can be a uid or a solid matrix.The particles interact via elastic repulsion, friction, adhesion and other surface forces

By nature, the length scales involved in these contact interactions are much smaller thanthe particle size External loading leads to particle deformations as well as cooperativeparticle rearrangements For instance, the particle deformations are of particular impor-tance in powder metallurgy but the particles may be considered as quasi-rigid beyondthe elastic deformation

From a macroscopic scale point of view, the phenomena that make granular als interesting are pattern formation, mixing or segregation, clustering (granular gases),avalanches, rotating ows, granular convection, and jamming/ unjamming For example,

materi-if a granular material is heaped on an inclined plane, then the large scale state of thesystem depends on the angle of the plane For large angles of this plane, the granularmaterial ows like a non-Newtonian liquid For small angles of this plane, the granularmaterial will behaves like a solid and remain stationary The critical value of the planeangle delimiting the two regimes (or phases) depends on the preparation history, and onthe transition between the phases The fundamental behavior of granular materials hasbeen widely studied in the literature [54] The following features have been investigated:(i) quasi-static deformation characteristics of granular materials at both small and largestrains, with discrete particles considered in the analysis;(ii) eects of mechanical pe-riodic excitation on a granular medium; (iii) advances in the constitutive modeling ofgranular materials using a continuum approach; (iv) interactions of particles at elevatedtemperatures, including adhesion forces and sintering; (v) critical state starting from dis-tinct initial conditions using experiments and computer simulations In summary, there

is a lot of interesting and complex physics to be understood by studying just the largescale properties of granular assemblies As the composition of the assembly becomesmore complicated, its behavior becomes even richer

From a meso-microscopic scale point of view, Radjạ and Lanier (in Cambou et

al [53]) noted that the geometrical changes of granular texture were at the origin of thecomplex rheology of granular materials These changes were highly nonlinear, involvingcreation and loss of contacts, rotation frustration and frictional sliding They depended

on the dissipative nature of contact interactions and steric exclusions among particles Inquasi-static deformation, various features of the plastic behavior such as shear strengthand dilatancy could be traced back to the evolution of the granular texture Therefore,these authors proposed to analyze and distinguish the variables as follows: granulartexture, kinematics, and force transmission in granular materials

The granular texture is disordered with many dierent variants depending on thecomposition (particle shapes and sizes), interactions and assembling procedure Thegranular texture evolves with loading The inhomogeneous distribution of contact forcesreects granular disorder in static equilibrium At the lowest order, the relevant scalarparameters concern the connectivity of this network At higher orders, the anisotropy

of the texture is described by fabric tensors The granular texture evolves mainly due

to contact losses and gains The fraction of lost and gained contacts depends on thecontact orientation During the deformation process, the gained contacts tend to move

in the direction of the major principal stress, conversely, the lost contacts are generallyobserved in the direction of the minor principal stress [55, 54]

The kinematics are directly linked to the deformation of granular media In otherwords, the kinematics are the displacement of particles Each grain has six degrees offreedom (three components of translation and three components of rotation) As anexample, the analysis of the displacement eld is shown in Fig 1.1 [56] A triaxialcompression test was carried out and combined with the X-ray microtomography to get

Figure 1.1: Particles displacement (color shows the displacement magnitude) duringtriaxial compression Results are shown as vertical slices Subgures from left to rightcorrespond to increasing vertical stresses (from Andò et al [56]).

the displacement eld within the sample

Figure 1.2: Photoelastic image of a small assembly of disks (from Dantu [57])

The contact forces are considered like the static variable of granular assemblies Theseforces appear when the granular materials are subjected to an external load The contactnetwork and the corresponding contact forces result from the reaction of the granularmaterial to the applied load and the transmission of internal forces The inhomogeneity

of these forces in granular assemblies were rst observed optically in assemblies of toelastic particles, which have the property to develop birefringence upon the application

pho-of stresses [57], as shown in Fig 1.2 This precursory experimental study allowed to serve the load bearing contacts and the establishment of force chains within a granularmaterial Since then, other studies have tried to use this technique to measure forces insuch granular assemblies [58] However, it is dicult to measure accurately tangentialforces experimentally

ob-Later, numerical simulations enabled to analyze and provide detailed evidence offorce chains, the classication of force networks in strong and weak networks, and theexponential distribution of strong forces [59,60] Moreover, the force probability density

function from simulation showed that the weak forces (below the average force) in asheared granular system have a nearly uniform or decreasing power-law shape which is

in agreement with experiments [61, 62]

1.1.2 Bonded granular materials

In general, two types of bonding are distinguished in the literature: solid bonding andliquid bonding While liquid bonding arises from the presence of capillary meniscii atthe grain contacts, solid bonds between grains may originate from dierent sources, such

as the process of sedimentation in natural soil and rock deposits or the precipitation of asolid, having either a natural or an articial origin During sedimentation, cementation isoften observed during the early diagenesis process [63] In the case of solid precipitation,

on the one hand, natural cementation takes various forms such as calcite, silica, ironoxides, or even clays [1,2,3,4,64,5] On the other hand, articial bonded materials areoften encountered in applications involving materials improved by mixing with Portlandcement or lime

Despite the complicatedness of the formation of the bonds, the eects of generalfeatures of solid bonding on the properties of granular materials were widely observedbased on experimental ndings: (i) the strength (dynamic and static) and small-strainstiness are enhanced [65, 66, 67, 68, 69]; (ii) the stress-strain and volumetric responsebecome relatively brittle and more dilative, respectively [67, 70, 71]; and (iii) quasi-preconsolidation pressure or yield stress can be observed in loading path [72, 73] Fur-thermore, the mechanical responses of cemented soils are found to depend on the amountand nature of the cementing agent [74] At the macroscopic scale, cementation induces

a strength enhancement and the occurrence of volumetric dilation [36, 37] In addition,

at the grain scale, the contact behavior of bonded materials was also observed by eral studies using the idealized granules bonded [75, 76], and the articially structuredbonds [77]

sev-Furthermore, liquid bonds were also observed in several experimental investigations.The eect of capillary bonding on the mechanical behavior of granular materials is ofprimary importance in powder technology [78,79,80] and transformations of geomateri-als [81,82,83] The eects of inter-grain cohesion were also observed in order to present

a general observation of the mechanical properties of wet granular materials [12, 34].Finally, the dynamics of wet granular matter were clearly reported in the study by Her-minghaus [84]

Besides, bonded granular materials have been extensively investigated using ical simulations in various contexts, such as: metallic powder processing [6, 85], mod-eling and treatment of ceramic powders [7, 8, 9] Assemblies of wet beads were ob-served [10,11,12], in which some microscopic observations are possible by combining nu-merical and experimental tests [11,12] A 2D numerical investigation into the microstruc-

numer-ture and mechanical properties of cohesive powders was carried out [13,14] Furthermore,the micro-macroscopic behavior of granular materials containing articial solid bridges,especially cemented sands, was extensively investigated [36, 37, 40,41, 86, 42,87, 88].

In addition, the eects of liquid bonds on the mechanical behavior of granular ter have been extensively studied in the past, and several models of capillary cohesionhave been proposed [89, 90, 91] The contribution of capillary bridges to the micro-macroscopic mechanical behavior has been investigated by means of numerical simula-tions [92, 24, 46,93, 94, 95, 29,47]

mat-The most relevant known features of bonded granular materials at the macroscale arethe yield criterion, the stress-strain response, the critical state, the inuence of cohesion,etc Here, we present the inuence of intergranular cohesion on the overall behavior.Macroscopic inuence of cohesion

A non-null cohesion in the Mohr-Coulomb failure criterion has an important sequence since it introduces a stress scale in the behavior of the material While non-cohesive materials are essentially sensitive to ratios of stress components, cohesive onesrespond dierently according to the absolute magnitude of stresses This is quite re-markable when dealing with compression tests in which all stress ratios remain constant.During an oedometric compression test, the vertical stress σ1 increases, while the ra-dial strain is maintained null Hence, the stresses are also observed to correspond tocoecient of lateral stress ratio σ3/σ1 In addition, it is well recognized that the den-sity of non-cohesive materials is slightly evolve with growing stresses, while cohesiveones undergo some irreversible compression under growing average pressure P In soilmechanics [2], the irreversible compression curves are usually described using a linearrelationship between the void ratio e (i.e e = −1 + 1/Φ with Φ = 1 − n, Φ is the solidfraction and n is the porosity) and the logarithm of P:

It was also applied to critical states, characterized by a specic value of the stress ratio

σ3/σ1, and the compaction curve is often modeled with the same slope Cc (called thecompression index) Relation (1.1) only applies for the monotonically growing pressures.The irreversibility of the plastic compression phenomenon implies that it does not apply

to pressure values lower than the highest pressure reached in the past, namely the consolidation pressure The reports on bonded materials are similar but the change ofdensity is considerably lower

pre-1.1.3 Wet granular materials

The mechanical properties of granular materials are remarkably aected when a smallamount of water is added For instance, a wet sand can be sculptured into quite stablestructures, for instance a sand castle, while this is impossible when it is dry However, thisphenomenon signicantly depends on the size of particles, as shown in Fig 1.3 (van derWaals attraction is computed for an inter-particle separation of 30 Å, the skeletal force

is shown for σ0 = 10 kPa and σ0 = 1 MPa) [96] These compressive forces mobilize theelectrical repulsion forces and bring particles together until compression and repulsion arebalanced Changing the pore uid can alter the inter-particle distance at equilibrium;the upper part of the gure shows the strain caused by changing the pore uid fromfresh-water to seawater concentrations

Figure 1.3: Skeletal versus contact-level capillary and electrical forces The upper part

of the gure shows the strain (axis on right) caused by changing the pore uid ionicconcentration from fresh-water to seawater conditions Note slopes: skeletal 2:1, weight3:1, capillary and van der Waals 1:1 (from Santamarina [96])

Here, several observations can be made as follows (from Santamarina [96]):

• Particle weight looses relevance with respect to capillary forces for particles smallerthan d ≈ 3 mm (point 1), and with respect to van der Waals attraction for particlessmaller than d ≈ 30 µm (point 2)

• Capillary forces can exceed the contribution of σ0

= 10 kPa connement for ticles smaller than d ≈ 20 µm (point 3) and the contribution of σ0 = 1 MPa for

par-d < 0.2 µm (point 4)

• Particles are considered coarse when skeletal forces due to boundary loads prevail.This is the case for particles larger than d ≈ 20 µm (point 3)

• Particles are fine when contact-level capillary and electrical forces gain relevance.This is the case when particles are smaller than d ≈ 1 − 10 µm.

Therefore, we can see that the smaller the particle size, the stronger the capillaryforces Even the moisture in the atmosphere may create tiny liquid bridges at the contactpoint between particles The presence of these liquid bridges induces capillary forces thatattract the particles to each other In reality, because of the asperities on the particlessurfaces, molecular interactions scale down to small adhesive forces, and capillary forcescan be large and dominant

Unlike in the nanoscale range of adhesive forces, they can apply from an interparticledistance of the order of the particle size Such unique properties of capillary interac-tions lead to remarkable changes in mechanical properties of granular materials in theappearance of the liquid in their interstitial spaces Therefore, the presence of liquidmenisci plays a key role in the overall mechanical properties of wet granular materi-als Based on the liquid content, wet granular materials can be classied in dierentregimes [97, 79,34], as given in Table 1.1

Table 1.1: Schematic diagrams for classication of wet granular materials with variousamount of liquid In the third column, the black circles represent the grains and the greyregions represent the interstitial liquid (from Mitarai and Nori [34])

• Pendular state: When a small amount of liquid is added to the granular material,the liquid initially collect near the contact point of particles In this regime, the

liquid content is limited to the domain where menisci can form without coalescence;

in this domain, the capillary forces are limited to pair-wise interactions

• Funicular state: As the liquid content is increased further, the neighbor liquidbridges start to coalesce The liquid lls some pores and multiple grains can be incontact with a given volume of liquid

• Capillary state: At higher values of liquid content most of the pores lls withliquid and large contiguous wet clusters forms

• Slurry state: In this regime all the pore space are fully saturated and the particlesare completely immersed in the liquid

In the rst three regimes, the role of capillary force is very important while it isnegligible in the slurry state

Many numerical and experimental studies reported that the presence of a smallamount of interstitial liquid strongly aects the yield stress of the materials The ef-fects of the presence of water on the apparent Coulomb cohesion was widely studied inthe literature Richefeu et al [24] studied the Coulomb cohesion in pendular state fordierent values of liquid content by using both numerical and experimental observations.They observed that the Coulomb cohesion increases nonlinearly with liquid content andsaturates to a maximum value, cm ' 600Pa at wm ' 0.03 They found a constant value

of friction angle ϕ (=tanư1µ), regardless the level of liquid content

In soil mechanics, Terzaghi [98] proposed the concept of eective stress to describethe mechanical properties of water-saturated soils Terzaghi's principle states that allmeasurable eects of a change of stress of the soil, that is compression, distortion andchange of shearing resistance, are exclusively due to changes in eective stress [99].Eective stress σ0

ij is dened as

with uw is the pore water pressure and the Kronecker's delta δij: δii = 1, δi6=j = 0

In the early works of Bishop [100], this principle was extended to unsaturated soils,as:

σij0 = (σij ư uaδij) + χ(uaư uw)δij, (1.3)where χ is called the eective stress parameter or Bishop's parameter, vary from 0 for drysoils to 1 for saturated soils, and uais the pore air pressure The terms σijưuaδij = σnetưijand uaưuw = s, dene the net stress and the matric suction, respectively Independently

of the stress state, the capillary forces are assumed to react in an isotropic way, generatingthe same pressure in all directions However, Scholtès et al reported on their recentstudies [93, 94] that this assumption is not valid These authors performed numericalsimulations of the triaxial compression of unsaturated granular materials They assumed

that the components of the stress tensor includes two terms: the contributions of contactforces σc

ij (eective stress), and an isotropic stress due to capillary interactions σcap

ij ,assuming an isotropic distribution of liquid bridges in the material The additive eect ofcapillary forces in stress tensor is veried for an initial isotropic conguration, althoughdeformation creates a slight anisotropy in liquid bridge distribution Therefore, thecontribution of capillary forces to the total stress is not an isotropic pressure, which isnot consistent with the Bishop form of the eective stress, or any expression ignoringthe deviatoric eect of the forces in liquid bridges

Moreover, at the microscopic level, the inuence of liquid bridges on the tensilestrength of unsaturated granular materials was observed in the pendular state by Kim

&Hwang [92] and Gröger et al [11] They numerically studied the tensile strength der isotropic pressure Kim and Hwang [92] measured the actual magnitude of tensilestrength induced by water in moist granular soils with low water contents (w < 4%).They found that the magnitudes of the measured tensile strength are signicantly dif-ferent from zero The tensile strength generally increases with increasing water contentand relative density They also proposed a model to estimate approximately the tensilestrength of an unsaturated granular material Gröger et al [11] reported that there was

un-a lun-arge inuence of the surfun-ace roughness on the tensile stresses in wet pun-article systems

By means of simulated tensile tests, they have shown that even when the tensile strength

is reached, elastic body contacts supporting the external tensile load still exist Theyalso simulated shear tests to compare with tensile test simulations and concluded thatfor a small tensile load, the yield locus appeared to be the straight extension of thegraphs for positive loads [11]

In addition, Gilabert et al [13] reported the inuences of assembling procedure ofmodel cohesive powders under low pressure They also studied the plastic compression,structural changes of this materials under isotropic loads with the same value of tensilestrength between grains [14] Under low pressure, the tensile strength plays a key role

to stabilize the loose structures Moreover, in recent study of Delenne et al [45], theynumerically investigated the process of growth and coalescence of liquid clusters in agranular material as the amount of liquid increases under the isotropic compaction Themost important nding of their work is that a peak grain pressure induced by capillaryforces in a granular packing occurs inside the funicular state It presents the transitionfrom a primary coalescence process, where the volume of the largest cluster remainssmall, to a secondary coalescence process governed by the increase of liquid clustervolumes carrying a larger capillary stress [45]

1.2 Experimental investigations

Aside from the conventional destructive techniques (e.g shear box, one-dimensionalcompression test, biaxial compression test, triaxial compression test, etc.), more and

more non-destructive methods have been widely applied to observe eectively the erties of granular materials, from macro- to microscopic level In order to have a generalview of non-destructive techniques, we present here a sketchy history of some recentadvanced techniques (type of imaging techniques) and its applications to granular mate-rials research, as given in Table 1.2 (following from Al-Raoush & Willson [101]) Thesetypical studies were carried out over the last thirty years (from 1982 to 2016) Thetechniques are briey presented and their application will be clearly illustrated in thefollowing sections.

Water content variations Sandy and ne sandy

loam

X-ray computed tomography * Crestana et al [103]

Water content and soil bulk

/den-sity

Welling-ton [105]Fractures, mud invasion, and

lithololgic characterization

Natural soil X-ray computed tomography * Hunt et al [106]

Air-lled porosity and pore size

distribution

Natural soil X-ray computed tomography * Warner et al [107]

Variation in water content Natural soil X-ray computed tomography * Cassel et al [108]

Topology and connectivity Fontainebleau

sand-stone

Synchrotron computed raphy

tomog-10 µm Spanne et al [109]

Porosity, volume fractions, and

specic surface area

Random soil samples Photoluminescence volumetric

imaging

Gray [110]Porosity, volume fractions,

pearmeability and connectivity

Fontainebleau stone

sand-X-ray computed tomography 7.5 µm Auzerais et al [111]

Porosity, specic surface area and

pore size distribution

Fontainebleau stone

sand-Synchrotron tomography 7.5 µm Coker et al [112]

Porosity, tortuosity, connectivity

and specic surface area

Bera sandstone, glassbead

X-ray computed tomography 5 µm Lindquist et al [113]

Pore size distribution,

coordina-tion, and specic surface area

Uniform glass beads Magnetic Resonance Imaging * Baldwin et al [114]

Pore size distribution Glass lter system Magnetic Resonance Imaging * Pauli et al [115]

with mercury porosimetry

400 µm Klobes et al [116]

Visualization of uid transport

through the sample

Sandy sediment soil Positron emission tomography * Khalili et al [117]

Porosity and water content Sandstone Synchrotron tomography 30 µm Coles et al [118]

Porosity and volume fractions Dolomite Gamma-ray computed

tomog-raphy

* Hsieh et al [119]

Hop-mans [120]Geometrical analysis Fontainebleau sand-

stone

Venkatarangan [121]Porosity, permeability, and spe-

cic surface area

Silty and clayeyquartz

Scanning electron microscopy 1.6 µm Solymar and Fabricius [122]

Critical angle of sandpiles Spherical glass beads Electron and uorescence